Paracontact Metric ðκ, μÞ-Manifold Satisfying the Miao-Tam Equation

Inspired by the positive mass theorem and the variational characterization of Einstein metrics on a closed manifold, with an aim to find a proper concept of metrics that would sit between constant scalar curvature metrics and Einstein metrics, in [1], Miao and Tam studied the variational properties of the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. Specifically, they derived the following sufficient and necessary condition for a metric to be a critical point:


Introduction
Inspired by the positive mass theorem and the variational characterization of Einstein metrics on a closed manifold, with an aim to find a proper concept of metrics that would sit between constant scalar curvature metrics and Einstein metrics, in [1], Miao and Tam studied the variational properties of the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. Specifically, they derived the following sufficient and necessary condition for a metric to be a critical point: Theorem 1 (Theorem 5 in [1]). Let Ω be a compact n -dimensional Riemannian manifold with smooth boundary Σ, γ be a given metric on Σ, and M K γ be the space of metrics on Ω which have constant scalar curvature K and have induced metric on Σ given by γ. Let g ∈ M K γ be a smooth metric such that the first Dirichlet eigenvalue of ðn − 1ÞΔ g + K is positive. Then, g is a critical point of the volume functional in M K γ if and only if there is a smooth function λ on Ω such that λ = 0 on Σ and where Δ g and ∇ 2 g are the Laplacian and Hessian operators with respect to g, and Ric(g) is the Ricci curvature of g.
For brevity, we call such critical metric as Miao-Tam critical metric and refer to equation (1) as the Miao-Tam equation. A fundamental property of a Miao-Tam critical metric is that its scalar curvature is a constant (see Theorem 7 in [1]). Some explicit examples of Miao-Tam critical metrics can be found in [1,2], including not only the standard metrics on geodesic balls in space forms but the spatial Schwarzschild metrics and AdS-Schwarzschild metrics restricted to certain domains containing their horizon and bounded by two spherically symmetric spheres. In [2], the authors classified all Einstein and conformally flat Miao-Tam critical metrics. In fact, they proved that any connected, compact, Einstein manifold with smooth boundary satisfying Miao-Tam critical condition is isometric to a geodesic ball in a simply connected space form. And then several generalizations of this rigidity result were found by different authors, replacing the Einstein assumption by a weaker condition such as harmonic Weyl tensor [3], parallel Ricci tensor [4], or cyclic parallel Ricci tensor [5]. For Some other generalizations or rigidity results, we can refer to [6][7][8][9][10], etc.
Recently, some geometricians focus on the study of Miao-Tam equation within the framework of contact metric manifolds. In [11], the authors proved that a complete K-contact metric satisfying the Miao-Tam critical condition is isometric to a unit sphere S 2n+1 . Furthermore, they studied ðk, μÞ-contact metrics satisfying the Miao-Tam equation.
Moreover, the Miao-Tam equation within the framework of Kenmotsu and almost Kenmotsu manifolds was studied in [12], and it was proved that a Kenmotsu metric satisfying the Miao-Tam equation is Einstein. In addition, in [13], the authors studied the critical point equation on K-paracontact manifolds; especially, they proved that any K-paracontact manifolds satisfying the Miao-Tam equation must be Einstein. We also note that some geometric structures such as Ricci soliton were studied within the framework of paracontact metric ðκ, μÞ-manifold (see [14]). In this direction, it is natural to study paracontact metric ðκ, μÞ-manifold satisfying the Miao-Tam equation. In this paper, we will prove the following main result:
A ð2n + 1Þ-dimensional smooth manifold M 2n+1 is said to have an almost paracontact structure ðφ, ξ, ηÞ, if it admits a ð1, 1Þ-tensor field φ, a vector field ξ, and a 1-form η satisfying the following conditions: The tensor field φ induces an almost paracomplex structure on each fiber of D = Ker ðηÞ, i.e., the eigendistributions D + and D − of φ corresponding to the eigenvalues 1 and −1, respectively, have same dimension n From the definition, it is easy to see that φξ = 0, η ∘ φ = 0, and the endomorphism φ have rank 2n. An almost paracontact structure is said to be normal if and only if the tensor field N φ ≔ ½φ, φ − 2dη ⊗ ξ vanishes identically. If an almost paracontact manifold admits a pseudo-Riemannian metric g such that for all X, Y ∈ ΓðTMÞ, then we say that M has an almost paracontact metric structure, and g is called compatible metric. It follows that η = gð·, ξÞ and gð·, φ · Þ = −gðφ · , · Þ. Notice that any such a pseudo-Riemannian metric is necessarily of signature ðn + 1, nÞ. If in addition dηðX, YÞ = gðX, φYÞ for all vector fields X, Y on M, then the manifold M 2n+1 ðφ, ξ, η, gÞ is said to be a paracontact metric manifold. In this case, η becomes a contact form, i.e., η ∧ ðdηÞ n ≠ 0, with ξ its Reeb vector field. In a paracontact metric manifold, one defines two self-adjoint opera-tors h and l by h = 1/2L ξ φ and l = Rð·, ξÞξ, where L ξ is the Lie derivative along ξ, and R is the curvature tensor of g. It is known in [25] that the two operators h and l satisfy And there also holds where ∇ is the Levi-Civita connection of the pseudo-Riemannian manifold ðM, gÞ. Moreover, h = 0 if and only if ξ is a Killing vector field, and in this case, the paracontact metric manifold M is said to be a K -paracontact manifold.
A normal paracontact metric manifold is said to be a paraSasakian manifold.
The study of nullity conditions on paracontact geometry is the most interesting topics in paracontact geometry. Motivated by the relationship between contact metric and paracontact geometry, in [18],. Cappelletti Montano et al. introduced the following.
Definition 3. A paracontact metric manifold M 2n+1 ðφ, ξ, η, gÞ is said to be a paracontact metric ðκ, μÞ -manifold, if its curvature tensor R satisfies for all tangent vector fields X, Y on M, where κ, μ are real constants.
Paracontact metric ðκ, μÞ-spaces satisfy (7) but this condition does not give any type of restriction over the value of κ, unlike in contact metric geometry, because the metric of a paracontact metric manifold is not positive definite. However, The geometric behavior of the paracontact metric ðκ, μÞ-manifold is different according κ < −1, κ = 1 and κ > −1. In particular, for the case κ < −1 and κ > −1, ðκ, μÞ-nullity condition (7) determines the whole curvature tensor field completely. The case κ = −1 is equivalent to h 2 = 0 but not to h = 0, which is different from contact ðκ, μÞ-space. Indeed, there are examples of paracontact metric ðκ, μÞ-spaces with h 2 = 0 but h=0, as was first shown in [18,27,28]. In this paper, we consider the paracontact metric ðκ, μÞ-manifolds with the condition κ > −1.

The Proof of Theorem 2
Before giving the proof of Theorem 2, we introduce some important lemmas which will be used later. First of all, we recall a basic fact about paracontact metric ðκ, μÞ-manifold.
In the following, we consider paracontact metric ðκ, μÞ -manifold satisfying the Miao-Tam equation.
Taking the covariant derivative of (12) along an arbitrary vector field Y on M, we obtain Similarly, we have for any vector field X, Y on M. Comparing the preceding two equations and using (12) in the well-known expression of the curvature tensor RðX, YÞ = ½∇ X , ∇ Y − ∇ ½X,Y , we obtain the result.

Lemma 6.
Let M 2n+1 ðφ, ξ, η, gÞ be a paracontact metric ðκ, μÞ -manifold of dimensional ð2n + 1Þ with κ > −1, and ðg, λÞ be a nonconstant solution of the Miao-Tam equation on M 2n+1 . Then, we have Proof. Firstly, taking covariant derivative of (8) along any vector field X, and using (4), we can obtain Taking the inner product of (10) with ξ and using (8) and (16), we have where f = −ðλS + 1Þ/ð2nÞ (noting that the dimension of M is 2n + 1). It follows from (6) that RðφX, φYÞξ = 0. Then, replacing X by φX and Y by φY in (17), respectively, we obtain Since λ is nonconstant on M, it is easy to see that Replacing X by φX in (9), we have Then, the action of h on the (20) gives where we have used (7). Operating (9) by φ, we have Replacing X by hX in (22) and using (7) again, we get Substituting equations (20)-(23) into (19) yields which completes the proof of Lemma 6. Next, we will give the complete proof of Theorem 2.

Advances in Mathematical Physics
Proof. Firstly, taking X = ξ in (17) gives Putting X = ξ in (6) and comparing with the forgoing equation, we obtain Noting that the scalar curvature S is a constant, it follows from f = −ðλS + 1Þ/ð2nÞ that Then, we can obtain from (26) and (27) that On the one hand, taking Y = ξ in (6), since hξ = 0, it follows that which gives Substituting (7) and (30) in (5), we get On the other hand, we obtain from (12) and (8) that Next, taking covariant derivative of (28) along ξ and making use of (31) and (32), we have Operating this equation by φ shows By the action of h in (34), it follows from (7) that Since we assume that κ > −1, we divide it into two cases: If case (i) occurs, it follows from Lemma 6 that κ = 0. Hence, the definition of paracontact metric ðκ, μÞ-manifold gives that RðX, YÞξ = 0 for any vector field X,Y. From Theorem 3.3 of [26], M 2n+1 is locally flat in dimension 3, and in higher dimensions (n > 1), it is locally isometric to the product of a flat ðn + 1Þ-dimensional manifold and an n-dimensional manifold of negative constant curvature −4.
This completes the proof of Theorem 2.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author(s) declare(s) that they have no conflicts of interest.